Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. == Motivation == In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map η : S 3 → S 2 , {\displaystyle \eta \colon S^{3}\to S^{2},} and proved that η {\displaystyle \eta } is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles η − 1 ( x ) , η − 1 ( y ) ⊂ S 3 {\displaystyle \eta ^{-1}(x),\eta ^{-1}(y)\subset S^{3}} is equal to 1, for any x ≠ y ∈ S 2 {\displaystyle x\neq y\in S^{2}} .